In this section, we review some general properties of boundary integrable quantum field theories (IQFTs) following the seminal paper of Ghoshal and Zamolodchikov [1]}. We consider two-dimensional Euclidean quantum field theories in flat spacetime with Cartesian coordinates \((x^1,x^2)=(x,y)\) . To quantize the system one has to choose the direction of time. We consider the theory defined on \((x,y) \in (s_\mathrm {L},s_\mathrm {R}) \times \mathbb {R}\) , where \(s_\mathrm {L}\)  and \(s_\mathrm {R}\)  denote the generic positions of the left and right boundary, respectively. In the so-called open channel, one chooses the \(y\)  direction as the direction of time. In this channel, the Hamiltonian reads
\( H_r=\int _{s_\mathrm {L}}^{s_\mathrm {R}}\mathcal {H}_{r}(\phi (x,y))\,\mathrm {d}x-\theta _{r\mathrm {L}}(\phi (s_L,y))+\theta _{r\mathrm {R}}(\phi (s_L,y)),\) 
